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Firstly, note that it is enough to construct an isomorphism $$\mathcal{C}(L(A,B),C) \simeq \mathcal{B}(B, R_2(A,C))$$ The third natural isomorphism then follows automatically. Secondly, standard application of Yoneda's lemma shows that if for each $A\in \mathcal{A}$ you have a right adjoint $R_2(A,\cdot)$ to $L(A,\cdot)$, then these right adjoints are natural in $A$ and assemble into a bifunctor $R_2(A,C)$. There are various standard theorems that can be used to verify existence of right adjoint to each $L(A,\cdot)$, like Special Adjoint Functor Theorem or Freyd's AFT. The exact statement depends on your categories and functors. The simplest statement is Freyd's AFT: $\mathcal{B}$ must be cocomplete and locally small, $L(A,\cdot)$ must preserve all small colimits and satisfy solution set condition.